# JAC Class 9 Maths Important Questions Chapter 1 Number Systems

Jharkhand Board JAC Class 9 Maths Important Questions Chapter 1 Number Systems Important Questions and Answers.

## JAC Board Class 9th Maths Important Questions Chapter 1 Number Systems

Question 1.
Find 4 rational numbers between 2 and 3.
Solution :
Write 2 and 3 multiplying in numerator and denominator with (4 + 1).
i.e. 2 = $$\frac{2 \times 5}{1 \times 5}=\frac{10}{5}$$
or 3 = $$\frac{3 \times 5}{1 \times 5}=\frac{15}{5}$$
So, the four required rational numbers are $$\frac {11}{5}$$, $$\frac {12}{5}$$, $$\frac {13}{5}$$, $$\frac {14}{5}$$

Question 2.
Find three rational numbers between a and b (a < b).
Solution :
a < b
⇒ a + a < b + a
⇒ 2a < a + b
⇒ a < $$\frac{a+b}{2}$$
Again, a < b
⇒ a + b < b + b
⇒ a + b < 2b
⇒ $$\frac{a+b}{2}$$ < b
∴ a < $$\frac{a+b}{2}$$ < b
i.e. $$\frac{a+b}{2}$$ lies between a and b.
Hence, 1st rational number between a and b is $$\frac{a+b}{2}$$

Question 3.
Find 6 rational numbers between $$\frac {1}{3}$$ and $$\frac {1}{2}$$
Solution :
LCM of 3 and 2 = 6
∴ $$\frac{1}{3}=\frac{2}{6}$$ and $$\frac{1}{3}=\frac{2}{6}$$
Now, $$\frac{2}{6}=\frac{20}{60}$$ and $$\frac{3}{6}=\frac{30}{60}$$
∴ $$\frac{21}{60}, \frac{22}{60}, \frac{23}{60}, \frac{24}{60}, \frac{25}{60}$$ and $$\frac {26}{60}$$ lie between $$\frac {1}{3}$$ and $$\frac {1}{2}$$.

Question 4.
Find 5 rational numbers between and $$\frac {3}{5}$$ and $$\frac {4}{5}$$
Solution :
As $$\frac{3}{5}=\frac{3 \times 10}{5 \times 10}=\frac{30}{50}$$ and
$$\frac{4}{5}=\frac{4 \times 10}{5 \times 10}=\frac{40}{50}$$
∴ 5 rationals between $$\frac {3}{5}$$ and $$\frac {4}{5}$$ are $$\frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}$$ and $$\frac {35}{50}$$.

Question 5.
Find two irrational numbers between 2 and 2.5.
Solution :
$$\sqrt{2 \times 2.5}=\sqrt{5}$$
Since there is no rational number whose square is 5. So, $$\sqrt{5}$$ is irrational.
Also, $$\sqrt{2 \times \sqrt{5}}$$ is an irrational number.
∴ Irrational numbers between 2 and 2.5 are $$\sqrt{5}$$ and $$\sqrt{3 \sqrt{5}}$$.

Question 6.
Find two irrational numbers between $$\sqrt{2}$$ and $$\sqrt{3}$$.
Solution :
$$\sqrt{\sqrt{2} \times \sqrt{3}}=\sqrt{\sqrt{6}}=\sqrt[4]{6}$$
Irrational number between $$\sqrt{2}$$ and $$\sqrt{3}$$ are $$\sqrt[4]{6}, \sqrt{\sqrt{2} \times \sqrt[4]{6}}=\sqrt[4]{2} \times \sqrt[8]{6}$$

Question 7.
Find two irrational numbers between 0.12 and 0.13.
Solution :
0.1201001000100001…….,
0.12101001000100001……

Question 8.
Find two irrational numbers between 0.3010010001……. and 0.3030030003…..
Solution :
0.302020020002……, 0.302030030003….

Question 9.
Arrange $$\sqrt{2}$$, $$\sqrt[3]{3}$$ and $$\sqrt[4]{5}$$ in ascending order.
Solution :
L.C.M. of 2, 3, 4 is 12.

Question 10.
Which is greater $$\sqrt{7}$$ – $$\sqrt{3}$$ or $$\sqrt{5}$$ – 1 ?
Solution :

⇒ $$\sqrt{7}$$ – $$\sqrt{3}$$ < $$\sqrt{5}$$ – 1 ⇒ $$\sqrt{5}$$ – 1 > $$\sqrt{7}$$ – $$\sqrt{3}$$
So, $$\sqrt{5}$$ – 1 is greater.

Question 11.
Find the R.F. (rationalizing factor) of the following :
(i) $$\sqrt{12}$$
(ii) $$\sqrt{162}$$
(iii) $$\sqrt[3]{4}$$
(iv) $$\sqrt[3]{16}$$
(v) $$\sqrt[4]{162}$$
(vi) 2 + $$\sqrt{3}$$
(vii) 7 – 4$$\sqrt{3}$$
(vii) 3$$\sqrt{3}$$ + 2$$\sqrt{2}$$
(ix) $$\sqrt[3]{3}$$ + $$\sqrt[3]{2}$$
Solution :
(i) $$\sqrt{12}$$
First write its simplest form i.e. 2$$\sqrt{3}$$.
Now R.F. of $$\sqrt{3}$$ is $$\sqrt{3}$$
∴ R.F. of $$\sqrt{12}$$ is $$\sqrt{3}$$.

(ii) $$\sqrt{162}$$
Simplest form of $$\sqrt{162}$$ is 9$$\sqrt{2}$$.
R.F. of $$\sqrt{2}$$ is $$\sqrt{2}$$.
∴ R.F. of $$\sqrt{162}$$ is $$\sqrt{2}$$

(iii) $$\sqrt[3]{4}$$
$$\sqrt[3]{4} \times \sqrt[3]{4^2}=\sqrt[3]{4^3}$$ = 4
∴ R.F. of $$\sqrt[3]{4}$$ is $$\sqrt[3]{4^2}$$ or $$\sqrt[3]{16}$$.

(iv) $$\sqrt[3]{16}$$
Simplest form of $$\sqrt[3]{16}$$ is 2$$\sqrt[3]{2}$$
Now, R.F. of $$\sqrt[3]{2}$$ is $$\sqrt[3]{2^2}$$
∴ R.F. of $$\sqrt[3]{16}$$ is $$\sqrt[3]{2^2}$$ or $$\sqrt[3]{4}$$.

(v) $$\sqrt[4]{162}$$
Simplest form of $$\sqrt[4]{162}$$ is 3$$\sqrt[4]{2}$$
Now, R.F. of $$\sqrt[4]{2}$$ is $$\sqrt[4]{2^3}$$ or $$\sqrt[4]{8}$$
R.E. of $$\sqrt[4]{162}$$ is $$\sqrt[4]{2^3}$$ or $$\sqrt[4]{8}$$

(vi) 2 + $$\sqrt{3}$$
As (2 + $$\sqrt{3}$$)(2 – $$\sqrt{3}$$) = (2)2 – ($$\sqrt{3}$$)2
= 4 – 3 = 1, which is rational.
∴ R.F. of 2 + $$\sqrt{3}$$ is 2 – $$\sqrt{3}$$.

(vii) 7 – 4$$\sqrt{3}$$
As (7 – 4$$\sqrt{3}$$)(7 + 4$$\sqrt{3}$$)
= (7)2 – (4$$\sqrt{3}$$)2 = 49 – 48
= 1, which is rational
∴ R.E. of 7 – 4$$\sqrt{3}$$ is 7 + 4$$\sqrt{3}$$.

(viii) 3$$\sqrt{3}$$ + 2$$\sqrt{2}$$
As (3$$\sqrt{3}$$ + 2$$\sqrt{2}$$)(3$$\sqrt{3}$$ – 2$$\sqrt{2}$$)
= (3$$\sqrt{3}$$)2 – (2$$\sqrt{2}$$)2 = 27 – 8
= 19, which is rational
∴ R.F. of 3$$\sqrt{3}$$ + 2$$\sqrt{2}$$ is 3$$\sqrt{3}$$ – 2$$\sqrt{2}$$

(ix) $$\sqrt[3]{3}$$ + $$\sqrt[3]{2}$$
As ($$\sqrt[3]{3}$$ + $$\sqrt[3]{2}$$)($$\sqrt[3]{3^2}$$ – $$\sqrt[3]{3}$$ × $$\sqrt[3]{2}$$ + $$\sqrt[3]{2^2}$$)
= ($$\sqrt[3]{3}$$)3 + ($$\sqrt[3]{2}$$)3 = 3 + 2
= 5, which is rational
∴ R.F. of $$\sqrt[3]{3}$$ + $$\sqrt[3]{2}$$
is ($$\sqrt[3]{3^2}$$ – $$\sqrt[3]{3}$$ × $$\sqrt[3]{2}$$ + $$\sqrt[3]{2^2}$$).

Question 12.
Express the following surd with a rational denominator.
$$\frac{8}{\sqrt{15}+1-\sqrt{5}-\sqrt{3}}$$
Solution :

Question 13.
Rationalise the denominator of $$\frac{a^2}{\sqrt{a^2+b^2}+b}$$
Solution :

Question 14.
If $$\frac{3+2 \sqrt{2}}{3-\sqrt{2}}$$ = a + b$$\sqrt{2}$$, where a and b are rationals then find the values of a and b.
Solution :

Equating the rational and irrational parts, we get
a = $$\frac {13}{7}$$ , b = $$\frac {9}{7}$$

Question 15.
If $$\sqrt{3}$$ = 1.732, find the value of $$\frac{1}{\sqrt{3}-1}$$.
Solution :

Question 16.
If $$\sqrt{5}$$ = 2.236 and $$\sqrt{2}$$ = 1.414, then evaluate : $$\frac{3}{\sqrt{5}+\sqrt{2}}+\frac{4}{\sqrt{5}-\sqrt{2}}$$
Solution :

Question 17.
If x = $$\frac{1}{2+\sqrt{3}}$$, find the value of x3 – x2 – 11x + 3.
Solution :
As, x = $$\frac{1}{2+\sqrt{3}}$$ = 2 – $$\sqrt{3}$$
⇒ x – 2 = –$$\sqrt{3}$$
⇒ (x – 2)2 = (-$$\sqrt{3}$$)2
[By Squaring both sides]
⇒ x2 + 4 – 4x = 3
⇒ x2 – 4x + 1 = 0
Now, x3 – x2 – 11x + 3
= x3 – 4x2 + x + 3x2 – 12x + 3
= x (x2 – 4x + 1) + 3 (x2 – 4x + 1)
= x(0) + 3 (0) = 0 + 0 = 0

Question 18.
If x = 3 – $$\sqrt{8}$$ , find the value of x3 + $$\frac{1}{x^3}$$.
Solution :

Question 19.
If x = 1 + 21/3 + 22/3, show that x3 – 3x2 – 3x – 1 = 0
Solution :
x = 1 + 21/3 + 22/3
⇒ x – 1 = (21/3 + 22/3)
⇒ (x – 1)3 = (21/3 + 22/3)3
⇒ (x – 1)3
⇒ (21/3)3 + (22/3)3 + 3.21/3 × 21/3(21/3 + 22/3)
⇒ (x – 1)3 = 2 + 22 + 3.21 (x – 1)
⇒ (x – 1)3 = 6 + 6(x – 1)
⇒ x3 – 3x2 + 3x – 1 = 6x
⇒ x3 – 3x2 – 3x – 1 = 0

Question 20.
Solve : $$\sqrt{x+3}+\sqrt{x-2}$$ = 5
Solution :
⇒ $$\sqrt{x+3}+\sqrt{x-2}$$ = 5
⇒ $$\sqrt{x+3}$$ = 5 – $$\sqrt{x-2}$$
⇒ ($$\sqrt{x+3}$$)2 = (5 – $$\sqrt{x-2}$$)2
[By squaring both sides]
⇒ x + 3 = 25 + (x – 2) – 10$$\sqrt{x-2}$$
⇒ x + 3 = 25 + x – 2 – 10$$\sqrt{x-2}$$
⇒ 3 – 23 = – 10$$\sqrt{x-2}$$
⇒ – 20 = – 10$$\sqrt{x-2}$$
⇒ 2 = $$\sqrt{x-2}$$
⇒ x – 2 = 4 [By squaring both sides]
⇒ x = 6

Question 21.
If x = 1 + $$\sqrt{2}$$ + $$\sqrt{3}$$, prove that x4 – 4x3 – 4x2 + 16 – 8 = 0.
Solution :
x = 1 + $$\sqrt{2}$$ + $$\sqrt{3}$$
⇒ x – 1 = $$\sqrt{2}$$ + $$\sqrt{3}$$
⇒ (x – 1)2 = ($$\sqrt{2}$$ + $$\sqrt{3}$$)2
[By squaring both sides]
⇒ x2 + 1 – 2x = 2 + 3 + 2$$\sqrt{6}$$
⇒ x2 – 2x – 4 = 2$$\sqrt{6}$$
⇒ (x2 – 2x – 4)2 = (2$$\sqrt{6}$$)2
⇒ x4 + 4x2 + 16 – 4x3 + 16x – 8x2 = 24
⇒ x4 – 4x3 – 4x2 + 16x + 16 – 24 = 0
⇒ x4 – 4x3 – 4x2 + 16x – 8 = 0

Question 22.
Evaluate each of the following:
(i) 52 × 54
(ii) 58 ÷ 53
(iii) (32)3
(iv) ($$\frac {11}{12}$$)3
(v) ($$\frac {3}{4}$$)-3
Solution :
Using the laws of indices, we have
(i) 52 × 54 = 52+4 = 56 = 15625
[∵ am × an = am+n]

(ii) 58 ÷ 53 = $$\frac{5^8}{5^3}$$ = 58-3 = 55 = 3125
[∵ am ÷ an = am-n]

(iii) (32)3 = 32×3 = 36 = 729
[∵ (am)n = am×n]

Question 23.
Evaluate each of the following.
(i) ($$\frac {2}{11}$$)4 × ($$\frac {11}{3}$$)2 × ($$\frac {3}{2}$$)3
(ii) ($$\frac {1}{2}$$)5 × ($$\frac {-2}{3}$$)4 × ($$\frac {3}{5}$$)-1
(iii) 255 × 260 – 297 × 218
(iv) ($$\frac {2}{3}$$)3 × ($$\frac {2}{5}$$)-3 × ($$\frac {3}{5}$$)2
Solution :
(i) We have,
($$\frac {2}{11}$$)4 × ($$\frac {11}{3}$$)2 × ($$\frac {3}{2}$$)3

Question 24.
Simplify :
(i) $$\frac{(25)^{3 / 2} \times(243)^{3 / 5}}{(16)^{5 / 4} \times(8)^{4 / 3}}$$
(ii) $$\frac{16 \times 2^{n+1}-4 \times 2^n}{16 \times 2^{n+2}-2 \times 2^{n+2}}$$
Solution :
We have

Question 25.
Simplify :
$$\left(\frac{81}{16}\right)^{-3 / 4} \times\left[\left(\frac{25}{9}\right)^{-3 / 2} \div\left(\frac{5}{2}\right)^{-3}\right]$$
Solution :

Multiple Choice Questions

Question 1.
If x = 3 + $$\sqrt{8}$$ and y = 3 – $$\sqrt{8}$$ then $$\frac{1}{x^2}+\frac{1}{y^2}$$ =
(a) – 34
(b) 34
(c) 12$$\sqrt{8}$$
(d) – 12$$\sqrt{8}$$
Solution :
(b) 34

Question 2.
If $$\frac{3+\sqrt{7}}{3-\sqrt{7}}$$ = a + b$$\sqrt{7}$$ then (a, b) =
(a) (8, -3)
(b) (-8, -3)
(c) (-8, 3)
(d) (8, 3)
Solution :
(d) (8, 3)

Question 3.
Which of the following is an irrational number ?
(a) 0.24
(b) $$0 . \overline{24}$$
(c) 0.5777….
(d) 0.242242224…
Solution :
(d) 0.242242224…

Question 4.
If x = $$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$$ and y = 1, then value of $$\frac{x-y}{x-3 y}$$ is :
(a) $$\frac{5}{\sqrt{5}-4}$$
(b) $$\frac{5}{\sqrt{6}+4}$$
(c) $$\frac{\sqrt{6}-4}{5}$$
(d) $$\frac{\sqrt{6}+4}{5}$$
Solution :
(d) $$\frac{\sqrt{6}+4}{5}$$

Question 5.
$$\sqrt{2}$$ is a/an _________ number.
(a) natural
(b) whole
(c) irrational
(d) integer
Solution :
(c) irrational

Question 6.
The value of $$\sqrt[5]{(32)^{-3}}$$ is:
(a) 1/8
(b) 1/16
(c) 1/32
(d) None
Solution :
(a) 1/8

Question 7.
If x = $$\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$$ and y = $$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$$ then value of x2 + xy + y2 is :
(a) 99
(b) 100
(c) 1
(d) 0
Solution :
(a) 99

Question 8.
Which of the following is not a rational number?
(a) 2 + $$\sqrt{3}$$
(b) – $$\frac {7}{11}$$
(c) 3.42
(d) $$0 .23 \overline{4}$$
Solution :
(a) 2 + $$\sqrt{3}$$

Question 9.
Which of the following is smallest?
(a) $$\sqrt[4]{5}$$
(b) $$\sqrt[5]{4}$$
(c) $$\sqrt{4}$$
(d) $$\sqrt{3}$$
Solution :
(b) $$\sqrt[5]{4}$$

Question 10.
The product of $$\sqrt{3}$$ and $$\sqrt[3]{5}$$ is:
(a) $$\sqrt[6]{375}$$
(b) $$\sqrt[6]{675}$$
(c) $$\sqrt[6]{575}$$
(d) $$\sqrt[6]{475}$$
Solution :
(b) $$\sqrt[6]{675}$$

Question 11.
The exponential form of $$\sqrt{\sqrt{2}} \times \sqrt{2} \times \sqrt{2}$$ is :
(a) 21/16
(b) 83/4
(c) 23/4
(d) 81/2
Solution :
(c) 23/4

Question 12.
The value of x, if 5x-3 × 32x-8 = 225, is:
(a) 1
(b) 2
(c) 3
(d) 5
Solution :
(d) 5

Question 13.
If 25x ÷ 2x = $$\sqrt[5]{2^{20}}$$ then x =
(a) 0
(b) – 1
(c) $$\frac {1}{2}$$
(d) 1
Solution :
(d) 1

Question 14.
If 10x = $$3 . \overline{3}$$ = 3 + x, then x =
(a) $$\frac {1}{9}$$
(b) $$\frac {1}{3}$$
(c) 3
(d) 9
Solution :
(b) $$\frac {1}{3}$$

Question 15.
A rational number between $$\frac {1}{7}$$ and $$\frac {1}{3}$$ is
(a) $$\frac {29}{210}$$
(b) $$\frac {50}{210}$$
(c) $$\frac {81}{210}$$
(d) $$\frac {93}{210}$$
Solution :
(b) $$\frac {50}{210}$$